Math 321 - Real Variables II - Spring 2015

Instructor: Malabika Pramanik
Office: 214 Mathematics Building
E-mail: malabika at math dot ubc dot ca
Lectures: Mon,Wed,Fri 9:00 AM to 10:00 AM in Room 104 Mathematics Building.
Office hours: Mon 10-11, Wed 11-12 or by appointment.

Practice problems

Here is a list of exercises from the textbook based on the weekly lecture material. They are not meant to be turned in for grading, but it is recommended that you work through them to get a better understanding of the topics covered. Some midterm and final exam problems may be modelled on these exercises. Unlike homework problems, detailed solutions of these problems will not be posted, but I am happy to include hints upon request.

• Sequences and series of functions
• Chapter 7: Exercises 1-26.For problem 22, assume that f is Riemann integrable instead of Riemann-Stieltjes integrable for now.
• More practice problems on Chapter 7.
• Riemann-Stieltjes integration
• Fourier series
• Chapter 8, Exercises 12-19.
• More exercises in the practice problem sheet for the final. See at the bottom of this page.

Real analysis lecture notes on the web

Here is a list of online lecture notes of similar courses offered at various institutions.

• MIT open courseware .
• John Lindsay Orr's Analysis Webnotes, University of Nebraska, Lincoln.
• Eric Sawyer's lecture notes ,McMaster University.
• Vern Paulsen's lecture notes, University of Houston.
• Lee Larson's lecture notes, University of Louiville.
• An introduction to real analysis by William Trench.

• Week-by-week course outline

Here is a rough guideline of the course structure, arranged by week. The textbook pages are mentioned as a reference and as a reading guide. The treatment of these topics in lecture may vary somewhat from that of the text.

• Week 1 (pages 143-154 of the textbook):
• Pointwise and uniform convergence
• Definitions and examples
• Interchanging limits.
• Week 2 (pages 143-154 of the textbook):
• Applications of pointwise and uniform convergence:
• Weierstrass M-test
• A continuous but nowhere differentiable function
• A space-filling curve
• Separability of the space of continuous functions on [0,1]
• Week 3 (pages 143-154 of the textbook):
• Density of polygonal functions in the space C[0,1]
• A quick review of uniform continuity
• Bernstein's proof of the Weierstrass approximation theorem
• Week 4 (pages 159-161 of the textbook):
• Approximation of continuous periodic functions by trigonometric polynomials
• Algebras and lattices
• The Stone-Weierstrass theorem - real version
• Week 5 (pages 154-165 of the textbook):
• Stone-Weierstrass theorem - complex version (a reading exercise)
• Review of compactness in general metric spaces
• Equicontinuity
• Statement of the Arzela-Ascoli theorem
• Week 6 (pages 120-130 of the textbook) :
• Proof of the Arzela-Ascoli theorem
• Towards Riemann-Stieltjes integral
• The Riemann-Stieltjes integral
• Riemann's condition for Riemann-Stieltjes integrability
• The space of Riemann-Stieltjes integrable functions
• Week 7 :
• Winter break.
• Week 8 :
• Riemann-Stieltjes integrability and uniform convergence
• Midterm review.
• Week 9 (pages 120-133 of the textbook):
• Functions of bounded variation
• Jordan's theorem
• General integrators
• Integration by parts
• Integrators of bounded variation
• Linear functionals on normed linear spaces
• Week 10 (pages 185-192 of the textbook) :
• Riesz representation theorem for continuous linear functionals on C[a,b]
• Fourier series
• Partial Fourier sums as L2 projections
• Week 11 (pages 185-192 of the textbook) :
• Bessel's inequality
• Plancherel's theorem
• L2 convergence of Fourier series
• L2 approximation by continuous periodic functions
• Dirichlet's formula
• Week 12 (pages 185-192 of the textbook):
• Properties of the Dirichlet kernel
• Uniform convergence, or lack thereof, of partial Fourier sums
• Fejer kernel and Cesaro summability of Fourier series
• A brief introduction to approximations to the identity
• Week 13 :
• Shortcomings of Riemann integration - some workarounds
• Sets of Lebesgue measure zero
• Lebesgue's condition on Riemann integrability

Midterm

• The midterm will be held in class on Friday February 27. The duration of the exam is 50 minutes.
• The syllabus includes the material covered in class up until Monday February 23. In particular, this covers all of Chapter 7 and a portion of Chapter 6 of the textbook.
• Here is the list of problems we discussed in class in Wednesday February 25, with a sketch of the solutions.
• The actual midterm with solutions.